Roceedings of Symposia In Pure Mathenutlca...

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Roceedings of Symposia In Pure Mathenutlca Volume 34. 1979

THE PROBABILITY OF AN INTERVAL GRAPH, AND WHY IT MATTERS

J o e l E. Cohen, Jdnos ~ o m l b s , and Thomas Mueller 1

ABSTRACT. An i n t e r v a l graph is t h e i n t e r s e c t i o n graph of a f ami ly

of i n t e r v a l s of t h e r e a l l i n e . I n t e r v a l graphs have been used f o r i n f e r ence

in s e v e r a l s c i ences , inc luding archeology, ecology, g e n e t i c s , and psychology.

I n t h e s e a p p l i c a t i o n s , t h e s t r e n g t h of i n f e r ence depends on t h e p r o b a b i l i t y

t h a t a random graph is an i n t e r v a l graph. Using a d e f i n i t i o n of a random

graph due t o Erdtls and ~ e ' n ~ i [ e l , we ob ta in exac t p r o b a b i l i t i e s and

asymptotic and Monte Car lo e s t ima te s of t h e p r o b a b i l i t i e s , f o r vary ing

numbers of v e r t i c e s and edges. We a l s o ob t a in t h e asymptot ic p r o b a b i l i t y

t h a t a randcm graph is a c i r c u l a r a r c graph, which is t h e i n t e r s e c t i o n

graph of a fami ly of a r c s on t h e c i r c l e . Some mathematical ques t i ons

a r i s i n g i n s c i e n t i f i c i n f e r ence remain unanswered.

1. INTRODUCTION. Since i n t e r v a l graphs were in t roduced 21 yea r s ago ,

they have been widely used i n s c i ence and mathematics. I n Sec t ion 2, we

review t h e d e f i n i t i o n and c h a r a c t e r i z a t i o n s of i n t e r v a l graphs. I n Sec t ion

3, we desc r ibe a p p l i c a t i o n s of i n t e r v a l graphs. Some of t he se a p p l i c a t i o n s

lead n a t u r a l l y t o t h e ques t ion , what i s t h e p r o b a b i l i t y t h a t a random graph

is an i n t e r v a l graph? I n Sec t ion 4, we d e f i n e a random graph. Using exac t

a n a l y s i s , asymptotic t heo ry , and Monte Carlo s imula t ion , we e s t ima te t h e

p r o b a b i l i t y t h a t a random graph is an i n t e r v a l graph. We a l s o f i n d t h e

asymptotic p r o b a b i l i t y t h a t a random graph is a c i r c u l a r a r c graph.

F i n a l l y , i n Sect ion 5, we p r e s e n t some problems which remain unsolved.

2. DEFINITION AND CHARACTERIZATIONS. Let G be a graph wi th v l a b e l l e d

v e r t i c e s a l, . . . , a and E un l abe l l ed , undi rec ted edges e v l, , eE.

Each edge may be w r i t t e n a s an unordered p a i r of v e r t i c e s (ai, a i )

= ( a a i ) , i # j , and t h e edge i s s a i d t o connect t h e v e r t i c e s a . and a j ' j '

AMS(M0S) s u b j e c t c l a s s i f i c a t i o n s (1970). Primary O5C30, 60C05; Secondary 55A15, 62E25, 92A05, 92A10.

'J.E.c. and T.M. were supported i n p a r t by g r a n t DEB(BMS) 74-13276 of t h e U, S, Nat iona l Science Foundation.

Copyright O 1979. American Mathematical Society

98 JOEL E . COHEN, J ~ N O S K O M L ~ S , A N D THOMAS MUELLER

Loops and mul t ip le edges a r e excluded. G i s an i n t e r v a l graph when the re

is a co l l ec t ion S 1'

. , S of open, c losed, or mixed i n t e r v a l s of t h e v

r e a l l i n e such t h a t the re is an edge between a and a i + j, i f and only i j' i f S. and S. overlap, t h a t is Si n S. + 0. Thus G is an i n t e r v a l graph i f

1 3 3 and only i f G i s t h e i n t e r s e c t i o n graph of some family of i n t e r v a l s of t h e

l i n e .

H is a subgraph of G i f t he v e r t i c e s of H a r e a subset of the

v e r t i c e s of G and t h e edges of H a r e a subset of t h e edges of G .

A subgraph H of G is an induced subgraph of G i f t h e r e is an edge

between two v e r t i c e s of H whenever t h e r e is an edge between those two

ve r t i ces i n G .

Lekkerkerker and Boland l141 cha rac te r i ze i n t e r v a l graphs when v i s

f i n i t e i n two ways.

F i r s t , G i s an i n t e r v a l graph i f and only i f it conta ins no induced

subgraph of t h e form pic tured i n Figure 1. (The graph I i n Figure 1

Fig. 1. The 5 graphs ( I , 1 1 ) o r c l a s s e s of graphs (111,' I V n , V ) forbidden n a s induced subgraphs according t o the cha rac te r i za t ion of Lekkerkerker and

Boland C141.

plays a s p e c i a l r o l e i n t h e asymptotic p robab i l i ty t h a t a random graph is

an i n t e r v a l graph. )

A reader who is in te res t ed only i n new r e s u l t s could now proceed

d i r e c t l y t o Section 4.

The second cha rac te r i za t ion r equ i re s t h r e e a d d i t i o n a l d e f i n i t i o n s .

A path i s an induced subgraph of G defined by a sequence of not necessar i ly

d i s t i n c t v e r t i c e s a 1~ • • • , a k such t h a t ( a a ) is an edge i n i' i + l

G, i = 1, . . . , k-1. An i r r e d u c i b l e pa th is a pa th i n which ai a j

f o r i + j and no v e r t e x i n t h e pa th is jo ined t o any v e r t e x i n t h e pa th o t h e r

than t h o s e immediately preceding o r f o l l owing it, when such e x i s t . A cyc l e

is any p a t h of t h e f o r m al, . . . , a k a al.

Then a graph G is an i n t e r v a l graph i f and only i f ( a ) G c o n t a i n s

no i r r e d u c i b l e c y c l e wi th more than t h r e e v e r t i c e s ; and ( b ) f o r any

t h r e e d i s t i n c t v e r t i c e s of G no two of which a r e joined by an edge, a t

l e a s t one of t h e v e r t i c e s is connected t o every pa th between t h e o t h e r two

v e r t i c e s .

Gilmore and Hoffman L l O l c h a r a c t e r i z e i n t e r v a l g raphs whether v is

f i n i t e or i n f i n i t e . Some f u r t h e r d e f i n i t i o n s a r e r equ i r ed . By a cyc l e ,

Gilmore and Hoffman mean any f i n i t e sequence of v e r t i c e s al, . . . a ak

of G such t h a t a l l of t h e edges (ai, a i t l ) , 1 I i I k-1 and (ak , a ) a r e 1 i n G and such t h a t if one t r a c e s ou t t h e g iven sequence of v e r t i c e s , one

does n o t t r a v e l a l ong t h e same edge twice i n t h e same d i r e c t i o n . Th i s

d e f i n i t i o n does n o t exclude v i s i t i n g a v e r t e x twice or t r a v e l i n g a long an

edge once i n each d i r e c t i o n . I f al, . . . , a k is t h e g iven sequence of

v e r t i c e s i n a cyc l e , t h e cyc l e is odd i f k is odd. A t r i a n g u l a r chord of

t h i s c y c l e is any one of t h e edges (ai, a , 1 I i I - 2 , a a ) o r i t 2 k - l a 1

(ak, a 2 ) . I f G i s any graph , t h e canplementary graph G' has t h e same

v e r t i c e s a s G b u t has an edge connec t ing two v e r t i c e s i f and only if t h a t

edge does no t occur i n G.

With t h e s e d e f i n i t i o n s , a graph G is an i n t e r v a l g raph i f and only

i f every q u a d r i l a t e r a l i n G has a d i agona l and every odd c y c l e i n G' has

a t r i a n g u l a r chord.

Fulkerson and Gross [91 g i v e a ma t r i x - t heo re t i c c h a r a c t e r i z a t i o n

af f i n i t e i n t e r v a l g raphs . Th i s c h a r a c t e r i z a t i o n has been t h e b a s i s o f

most machine ccmputat ion i nvo lv ing i n t e r v a l g raphs , i nc lud ing t h e Monte

Ca r lo r e s u l t s de sc r i bed below.

Again, some d e f i n i t i o n s a r e needed. A c l i q u e of a graph G is a

subgraph of G which is complete, t h a t is , i n which every p a i r of v e r t i c e s

is jo ined by a n edge. If t h e f a m i l y of subgraphs of G which a r e c l i q u e s

i s p a r t i a l l y o rde r ed by s e t i nc lu s ion , t h e maximal elements of t h e p a r t i a l

a rde r i ng a r e c a l l e d dominant c l i q u e s . The dominant c l i q u e ve r su s v e r t e x

ma t r i x of G is a m a t r i x with one row f o r each dominant c l i q u e and one column

f o r each v e r t e x of G. The element i n t h e i t h row and j t h column is 1 i f

t h e j t h v e r t e x is a v e r t e x of t h e i t h dominant c l i q u e , and is 0 o therwise .

Since two v e r t i c e s of G a r e jo ined by an edge if and only i f t hey a r e

bo th v e r t i c e s i n s ane dominant c l i q u e , t h e dominant c l i q u e ve r su s v e r t e x

ma t r i x s p e c i f i e s G un ique ly , and v i c e ve r s a . A (0.1) ma t r i x has t h e

100 JOEL E. COHEN, JANOS K Q M L ~ S , AND THOMAS MUELLER

c o n s e c u t i v e 1 's p r o p e r t y i f and on ly i f t h e r e i s some permuta t ion of t h e

rows a f t e r which t h e 1's i n each column occur c o n s e c u t i v e l y . Thus a m a t r i x

has t h e c o n s e c u t i v e 1 ' s p r o p e r t y i f some ( p o s s i b l y n u l l ) r e o r d e r i n g of t h e

rows r e s u l t s i n no two 1's i n a g i v e n column being s e p a r a t e d by a 0.

Fulkerson and Gross g i v e a n e x p l i c i t a l g o r i t h m f o r t e s t i n g whether a ( 0 , l )

m a t r i x has t h e c o n s e c u t i v e 1's p r o p e r t y .

Then G i s a n i n t e r v a l g raph i f and o n l y i f t h e dominant c l i q u e v e r s u s

v e r t i x m a t r i x of G h a s t h e c o n s e c u t i v e 1 's p r o p e r t y .

3. APPLICATIOIiS. According t o Llerge C2J, ~ a j b s L l l ] o r i g i n a t e d

i n t e r v a l g r a p h s wi th t h e f o l l o w i n g , a p p a r e n t l y h y p o t h e t i c a l , problem:

Suppose each s t u d e n t a t a u n i v e r s i t y v i s i t s t h e l i b r a r y f o r e x a c t l y one

i n t e r v a l d u r i n g t h e d a y , and r e p o r t s a t t h e end of t h e day t h e o t h e r

s t u d e n t s who were t h e r e whi le he was. If each v e r t e x of t h e g r a p h G

cor responas t o one s t u d e n t , and two v e r t i c e s a r e j o i n e d by a n edge if and

on ly i f t h e two cor responding s t u d e n t s were i n t h e l i b r a r y s i m u l t a n e o u s l y ,

t h e n G is a n i n t e r v a l graph. The problem i s t o c h a r a c t e r i z e wnich graphs

could o r cou ld n o t a r i s e frcm such a p r o c e s s ,

Independent ly of HajGs, Benzer L1J posed t h e f o r m a l l y i d e n t i c a l

problem of d e c i d i n g whether , a t t h e l e v e l of g e n e t i c f i n e s t r u c t u r e ,

muta t ions i n t h e rII r e g i o n of t h e v i r u s , b a c t e r i o p h a g e T4, a r e l i n k e d

t o g e t h e r i n a l i n e a r s t r u c t u r e . Recombination exper iments can i n d i c a t e

whether any two mutant r e g i o n s o v e r l a p . If each v e r t e x of a graph G

cor responds t o one mutant and two v e r t i c e s a r e jo ined by an eage i f and o n l y

i f t h e two cor responding mutant r e g i o n s o v e r l a p , t h e n t h e g e n e t i c f i n e

s t r u c t u r e i s compat ib le w i t h a l i n e a r o r d e r if and on ly if G i s an i n t e r v a l

graph. For complete o v e r l a p d a t a on 19 mutan ts and incomple te d a t a on a

t o t a l of 145 mutan ts of T4, a l i n e a r model is a d e q u a t e . E l e c t r o n

microscopic and a u t o r a d i o g r a p h i c p i c t u r e s of t h e g e n e t i c m a t e r i a l of T4

confirm i t s p h y s i c a l l i n e a r i t y .

Benzer L11 i n v e s t i g a t e s t h e p r o b a b i l i t y t h a t randomly g e n e r a t e d d a t a

would b e compat ib le w i t h a l i n e a r one-dimensional s t r u c t u r e and o b t a i n s a n

upper bound. He r a i s e s t h e impor tan t q u e s t i o n of whether h i s d a t a c o u l d

d i s c r i m i n a t e a l i n e a r one-dimensional s t r u c t u r e from a p l a u s i b l e a l t e r n a t i v e ,

e . g . b r a n c h e d , s t r u c t u r e . T h i s b r i l l i a n t paper remains worth r e a d i n g a s

a s o u r c e of mathemat ica l problems and b i o l o g i c a l i d e a s .

Rober t s ~ 1 6 1 rev iews a p p l i c a t i o n s of i n t e r v a l g raphs t o t h e

p s y c h o l o g i c a l t h e o r y o f p r e f e r e n c e , a rcheo logy , developmental psychology, and

t h e t iming of t r a f f i c l i g h t s .

THE PROBABILITY OF AN INTERVAL GRAPH, AND WHY IT MATTERS 101

I n a preference experiment, a person is asked whether he p re fe r s one

wine, f o r example, t o another or is ind i f fe ren t between them. I f each wine

corresponds t o a ver tex of a graph G , and two ver t i ces a r e joined i f and only

if the individual is ind i f fe ren t between the corresponding wines, then G is

an i n t e r v a l graph i f and only i f t h e r e is some l i n e a r ordering along which

each wine can be assigned t o an in te rva l . See a l s o the re la ted work of

Hubert L131.

In archeology, the problem is t o es tab l i sh a se r i a t ion , or

chronological order, f o r graves or s t r a t a on the bas i s of the a r t i f a c t s found

in them. Each a r t i f a c t is c l a s s i f i e d i n t o one of a f i n i t e number of s t y l e s .

It is supposed t h a t each s t y l e of a r t i f a c t was put i n graves during one time

in te rva l . It is a l s o supposed t h a t i f a r t i f a c t s of two d i f f e r e n t s t y l e s

a r e found together i n a grave, then the time in te rva l s during which they were

made overlapped. The graph G with ve r t i ces corresponding t o s t y l e s , and

edges between every p a i r of s t y l e s found i n a common grave, is an i n t e r v a l

graph if and only i f some assignment of a time i n t e r v a l t o each s t y l e is

possible.

In the app l ica t ion of i n t e r v a l graphs t o developmental psychology,

it is assumed t h a t t r a i t s a r i s e i n a l l chi ldren i n a s ing le sequence

with temporal overlapping among t r a i t s . Children a r e assessed f o r t h e

simultaneous presence of various t r a i t s . I f the ve r t i ces of a graph G

correspond t o t h e t r a i t s and a r e joined by an edge i f the corresponding

t r a i t s appear simultaneously i n some ch i ld , then G is an i n t e r v a l graph

if and only i f it is possible t o assign some time i n t e r v a l t o each t r a i t .

In timing t r a f f i c l i g h t s , t h e problem is: given a graph G with

ve r t i ces corresponding t o the streams of t r a f f i c a t an in te r sec t ion ,

and edges between two v e r t i c e s i f the corresponding streams could sa fe ly

be permitted t o flow a t t h e same time, G is an i n t e r v a l graph i f and only

if each t r a f f i c stream i s permitted t o flow during a s ingle time i n t e r v a l i n

one cycle of the t r a f f i c l i g h t s .

Though many of these substantive appl icat ions requ i re spec ia l

extensions and refinements of theory, discussed by Roberts [16], the

concept of in te rva l graphs captures the l i o n ' s share of t h e formal s t ructure .

Booth and Lueker [41 and Booth C31 review appl icat ions of i n t e r v a l

graphs t o the assignment of records t o t r acks on a computer's disk memory fo r

e f f i c i e n t information r e t r i e v a l , Gaussian elimination schemes f o r sparse

symmetric posi t ive d e f i n i t e matrices (see a l s o Tarjan [17]), and table-driven

parsing. In te rva l graphs have been used t o determine the f e a s i b i l i t y of

proposed schedules of t a sks i n the building of Large ships (Alan J. Hoffman,

personal communication, 20 March 1978 ).

JOEL E . COHEN, J ~ N O S K O M L ~ S , AND THOMAS MUELLER

An app l i ca t ion of i n t e r v a l graphs t o ecology leads na tu ra l ly t o

t h e problem of determining t h e p robab i l i ty t h a t a random graph i s an i n t e r v a l

graph (Cohen C51, C61, 171).

I n ecology, a food web W i s a d i r ec ted graph t h a t t e l l s which kinds

of organisms nourish which o the r kinds of organisms i n a community of

species. Each l a b e l l e d ver tex i n W corresponds t o a kind of organism.

Each arrow o r d i r ec ted edge (ai, a . ) from ver tex i t o ver tex j s p e c i f i e s I

a flow of energy o r b ianass (food, i n s h o r t ) from the i t h kind of organism

t o t h e j t h kind of organism.

Another descr ip t ion of communities of species r ep resen t s each kind

of organism by a multidimensional hypervolume i n a hypothet ica l ecological

niche space. I n niche space, each dimension corresponds t o some environmental

va r i ab le or some va r i ab le character iz ing t h e food consumed by the organisms.

The multidimensional hypervolume associa ted with each kind of organism is

cal led its niche. The projec t ion of ecological niche space onto the

dimensions cha rac te r i z ing t h e food consumed is ca l l ed t roph ic niche space,

and t h e same projec t ion of a niche i s ca l l ed the t roph ic niche.

An elementary question about niche space is: what is t h e minimum

dimensionali ty of a niche space necessary t o r ep resen t , or t o descr ibe

completely, t h e overlaps among observed niches ? This question remains

unanswered f o r niche space i n genera l . The use of i n t e r v a l graphs g ives

a p a r t i a l answer f o r t roph ic niche space.

We de f ine t h e predators i n a food web graph W a s the s e t of a l l

kinds of organisms which consume some kind of organism i n W, o r

more formally a s t h e s e t of v e r t i c e s a such t h a t ( a a . ) is an arrow j i 3

in W, f o r some ai. Cannibalism is n o t excluded. Prey a r e defined a s t h e

s e t of a l l kinds of organisms t h a t a r e consumed by some kind of

organism, i . e . , a l l ai such t h a t ( a . , a . ) i s an arrow in W, f o r some vertex 1 3

a The t roph ic niche overlap graph G(W) is defined a s an undirected j '

graph i n which the v e r t i c e s a r e the predators i n a food web graph W. Two

predators a r e joined by an undirected edge when t h e r e is some kind of

prey t h a t both p reda to r s e a t . That is, ( a j , a k ) = (ak , a . ) is an edge 3

i n G(W) i f and only i f t h e r e e x i s t s sane ai i n W such t h a t both (ai, a . ) 3

and (ai, a k ) a r e arrows i n W.

I f t h e t roph ic niche of a kind of organism is a connected region

i n t roph ic niche space, then it is poss ib le f o r t roph ic niche overlaps

t o be described i n a one-dimensional space i f and only i f t h e t rophic

niche overlap graph G(W) i s an i n t e r v a l graph.

An ana lys i s of t h e niche overlap graphs of 30 r e a l food webs suggests

t h a t a niche space of dimension one s u f f i c e s t o descr ibe t h e t rophic

niche overlaps implied by r e a l focd webs i n s ing le h a b i t a t s .

To determine whether chance alone might explain why observed

t roph ic niche over lap graphs a r e i n t e r v a l graphs, it is necessary t o g i v e

p rec i se meaning t o "chance alone." Seven p r o b a b i l i s t i c models a r e described

and compared with d a t a i n Cohen L73. Six of t hese a r e models of focd web

graphs W; associa ted wi th each such model i s an induced p robab i l i t y

d i s t r i b u t i o n on G(W). One model a s s igns a p robab i l i t y d i s t r i b u t i o n d i r e c t l y

t o niche overlap graphs G(W). I n t h e next sec t ion we s h a l l s tudy t h e

p robab i l i t y t h a t a random graph G is an i n t e r v a l graph under t h i s model.

We brought t h e problem of enumerating i n t e r v a l graphs t o the a t t e n t i o n

of Frank Harary (Harary and Palmer [123), but t o our knowledge t h e r e have been

no r e s u l t s o the r than those we present here.

The p r o b a b i l i t i e s we c a l c u l a t e a r e r e l evan t not only t o ecology,

but t o any s c i e n t i f i c app l i ca t ion where i n t e r v a l graphs a r e used f o r

inference about t h e dimensionali ty of some hypothet ica l underlying s t r u c t u r e .

4. PROBABILITY THAT A RANDOM GRAPH IS AN INTERVAL GRAPH. The model

of a random graph which we s h a l l s tudy i s t h a t defined by Erdbs and

~ 6 n y i [ e l . Their systematic i nves t iga t ion of t h e p rope r t i e s and s t r u c t u r e s

of random graphs is a bas i c source of r e s u l t s and methods i n t h i s a rea .

They de f ine a random graph with v l abe l l ed v e r t i c e s and E edges a s one i n

which t h e E edges a r e chosen randomly (without replacement) among t h e

(;I poss ib l e edges s o t h a t a l l C poss ib l e graphs a r e equiprobable, where v,E

We a l s o use an equivalent cons t ruc t ion of a random graph with E

edges on v l abe l l ed v e r t i c e s . I f k edges a r e a l r eady chosen, choose one v

of t h e remaining ( ) - k edges, each with equal p robab i l i t y l/[(;)-kl, 2 k = 0, 1, .. . , E-1.

I f A is a proper ty which a graph e i t h e r possesses o r does no t

possess, and A i s t h e number of graphs on v l abe l l ed v e r t i c e s and E v,E

edges which possess t h e property, then the p robab i l i t y P (A) t h a t a v,E

random graph has property A i s defined a s AV,E/CV,E.

Let T be the event t h a t a graph i s an i n t e r v a l graph and -T be

t h e event t h a t a graph i s not an i n t e r v a l graph. We f i r s t obta in t h e

exact p robab i l i t y P (T) t h a t a random graph with v l abe l l ed v e r t i c e s v,E

and E edges is an i n t e r v a l graph f a r a l imi ted range of values of v and E.

Then we develop some asymptotic r e s u l t s . F i n a l l y , we at tempt t o connect

the two kinds of r e s u l t s with Monte Carlo s imula t ions .

J O E L E . COHEN, J ~ N O S K O M L ~ S , AND THOMAS MUELLER

v THEOREM 1. A l l graphs with E < 4 o r E = ( 2 ) - k, k = 0 o r 1, a r e

i n t e r v a l . For v > 3 ,

( i ) p V s 4 ( ~ ) = 1 - YB/c (V~-V-~) (V+Z) (V+~) I 2 2 ( i i ) PV,,(T) = 1 - 2 4 0 ( ~ - 4 ) ( ~ + ( 2 3 / 5 ) ) / C ( ~ - V - ~ ) ( V + ~ ) ( V + ~ ) ( V -v-8)l

3 2 ( i i i ) P (T) = 1 - (v-4)(720v +11424v -122064~+261600)/[(v+l)(v+2)

v,6 2 2 *(v2-v-4)(v - v - ~ ) ( v -v-1011

PROOF. ( i ) A graph with 4 edges can f a i l t o be i n t e r v a l i f and only i f

t h e 4 edges form a q u a d r i l a t e r a l . We can th ink of p lac ing a q u a d r i l a t e r a l

on v v e r t i c e s a s f i r s t placing a complete subgraph (with 6 edges) on 4

v e r t i c e s ( i n (El pos s ib l e ways) and then d e l e t i n g from each such complete

subgraph 2 edges s o t h a t t h e remaining 4 farm a q u a d r i l a t e r a l ( i n 3

poss ib l e ways). The t o t a l number of pos s ib l e ways of p lac ing t h e 4 edges

is Cv,4. Thus P (-TI = 3 ( ; ) / ~ ~ , ~ , from which t h e given P (T) fo l lows. v ,4 v*4

( i i ) F igure 2 g ives t h e 3 ways i n which 5 edges may be placed on

5 o r 6 v e r t i c e s t o form a graph which i s not an i n t e r v a l graph. For any

Fig. 2. The 3 ways 5 edges may be placed on 5 ar 6 v e r t i c e s t o form a graph

which is not an i n t e r v a l graph.

s e t of 5 l a b e l l e d v e r t i c e s , p a t t e r n ( a ) can be chosen i n 5 x 4 x 3 = 60

ways. For any s e t af 6 l a b e l l e d v e r t i c e s , p a t t e r n ( b ) can be chosen i n

15 x 3 = 45 ways. For any s e t of 5 l a b e l l e d v e r t i c e s , p a t t e r n ( c ) can

be chosen i n 12 poss ib l e ways. Thus P (-TI = [60(;) + 12(;) + 4 5 ( : ) 1 / ~ ~ , ~ . v , 5

( i i i ) F igure 3 g i v e s 11 of tlze 12 ways 6 edges may be placed on 5 o r

more v e r t i c e s t o form a graph which is not an i n t e r v a l graph. For each

p a t t e r n with n l a b e l l e d v e r t i c e s , n = 5, 6, 7, 8, t h e r e a r e ('1 choices of

t h e v e r t i c e s . Elementary counting shows t h a t t h e number of ways of 5 ass igning 6 edges t o form each p a t t e r n is: ( a ) 60 = 5 x 4 x 3; ( b ) 10 = ( 2 ) ;

6 ( c ) 6 0 = 6!/(2x6); ( d ) 360 = 12 x 5 x 6; ( e ) 360 = ( ) x 2 x 3 x 4; 2 ( f ) 180 = 6 ! / ( 2 ~ 2 ) ; (g , which is i d e n t i c a l t o graph I i n Figure 1 ) 840 (see

Fig. 3. Eleven of t h e 12 ways 6 edges may be placed on 5 o r more v e r t i c e s

t o form a graph which i s not an i n t e r v a l graph.

7 7 proof of Theorem 3 below); (h) 252 = 1 2 ~ ( ~ ) ; ( i ) 1260 = 60x( ); ( j ) 315

7 8 2 = (4)x3x3; (k) 630 = ( 4 ) ~ 3 ~ 3 . The twe l f th p a t t e r n , i d e n t i c a l t o graph IV - 2

b i n Figure 1, may be chosen i n 120 = (3)X3! ways. Multiplying each

v of these numbers by t h e appropr ia te ( ), summing, and dividing by C g ives n v,6

P (-TI, and co l l ec t ing powers of v y ie lds ( i i i ) . The f i n a l r e s u l t shows v,6

co r rec t ly t h a t P (T) = 1. 4,6 v

( i v ) A graph with ( 2 ) - 3 edges may be viewed as a complete graph

frcm which 3 edges have been se lec ted f o r omission. I f these 3 edges

connect a t o t a l of 3 v e r t i c e s , t h e r e is exact ly one way the 3 edges can be

chosen f o r each s e t of 3 v e r t i c e s ; each graph obtained f r o m a complete

graph by omitt ing such a t r i a n g l e i s an i n t e r v a l graph. I f t he 3 cmitted

edges involve exact ly 4 po in t s , t h e 3 omitted edges must c o n s t i t u t e a

p a t t e r n of t h e form (al, a2 IS (alS a3), (al, a4) (which can be chosen i n 6

16 = (,I - 4 ways) i n order f o r the r e s u l t t o be an i n t e r v a l graph. Thus " v v

P - 3 ) = [16(4) + ( 3 ) I / C v , ( ~ ) - 3 . 2

( v ) To obta in an i n t e r v a l graph by omitt ing 2 edges from a complete

graph, t h e 2 edges must form a pa t t e rn l i k e (al, a2 ) , (alS a3 1. For each

s e t of 3 l abe l l ed v e r t i c e s t h e r e a r e 3 ways of choosing such a pa t t e rn .

Thus P v (1) = 3(;)/$ (v)-? = 4/(v+l) . This proves Theorem 1. v, (2)-2 ' 2

JOEL E. COHEN, JANOS K O M L ~ S , AND THOMAS MUELLER

Theorem 1 covers a l l pos s ib l e numbers E of edges only f o r graphs wi th

up t o v = 5 v e r t i c e s .

To c a r r y out t h e asymptotic ana lys i s , we r e c a l l some more concepts

and r e s u l t s frcm b d B s and R6nyi La]. f ( v ) is c a l l e d a threshold func t ion f o r t h e proper ty A i f f o r any

E > 0 t h e r e a r e p o s i t i v e numbers 6, A, and v o such t h a t f o r v > vo,

E I 6 f ( v ) we have P (A) < E, and f o r E 2 Af(v) we have P (A) > 1 - E. v,E v,E

I f a graph G has v v e r t i c e s and E edges, t h e degree d of t h e

graph is 2E/v, which is t h e average degree of t h e v e r t i c e s of G. G

is s a i d t o be balanced i f no subgraph H 'of G has a l a r g e r degree than G

i t s e l f .

THEOREM A (Erdus and R6nyi [El, p. 231). Let v 2 2 and E be p o s i t i v e

i n t ege r s . Let B denote an a r b i t r a r y non-empty c l a s s of connected balanced v,E

graphs wi th v v e r t i c e s and E edges. The threshold func t ion f o r t h e proper ty

t h a t a random graph on n l a b e l l e d v e r t i c e s con ta ins a t l e a s t one subgraph

isomorphic wi th scme element of B is n 2-v/E - 2-2/d - n , where d i s t h e degree v,E

of each graph i n BV,E.

Here we g ive without proof a s l i g h t gene ra l i za t ion of Theorem A

which can be obta ined by us ing t h e inc lus ion-exclus ion formula and

c a l c u l a t i o n s a long t h e l i n e s of [El].

We say t h a t a graph G i s s t r o n g l y balanced i f any subgraph of G has

a smal ler degree than G.

Let B be an a r b i t r a r y f i n i t e c l a s s of s t r o n g l y balanced graphs

G1, . . . , G a l l having t h e same degree d. Let t h e number of ( l a b e l l e d ) m v e r t i c e s of G . be v and t h e number of edges be E.. Let B . denote t h e number

1 i 1 1

of graphs wi th v l a b e l l e d v e r t i c e s which a r e isomorphic t o G i is

THEOREM 2. Let A denote t h e event t h a t a random graph con ta ins k

exac t ly k subgraphs each iscmorphic t o some element of B. Assume t h a t , a s t h e

number v of l a b e l l e d v e r t i c e s of a random graph is increased , t he number

E(v) of edges is a l s o increased s o t h a t l i m - E(v)/v? ' 'Id = c . Here d is

t h e degree of each graph i n B, while E(v) and v r e f e r t o t h e edges and

v e r t i c e s of t h e randcm graph. Then P (A ), k = 0, 1, 2 , . . . , i s , v,E(v) k

asymptot ica l ly , a Poisson d i s t r i b u t i o n

where we de f ine p = E(v)/(;) and

I n p a r t i c u l a r ,

A i s t h e asympto t ic expected number of subgraphs isomorphic t o some graph

G . i n B. I n t h e a l t e r n a t i v e model ( c a l l e d r** i n [8, p. 201) of a randcm 1 v,E

graph i n which each edge i s chosen independently accord ing t o a Be rnou l l i

t r i a l with p r o b a b i l i t y p o f success , A is t h e p r e c i s e expected number of

subgraphs isomorphic t o scme graph Gi i n B. A s v i n c r e a s e s , A + A* where

and we have t h e p r e c i s e s ta tement

l i m P (A 1 = ( ~ * ) ~ e " * / k ! . rroJ v,E(v) k

THEOREM 3. Assume t h a t l i m - E ( v ) / v ' / ~ = c . Then l i m Pv,E(v) ,A+;

(T) 6 -A = e , where A?r = 32c /3. Furthermore, P (TI .. e f o r l a r g e v and E

6 5 v,E 6 6 5 a s long a s E /v i s no t t o o l a r g e , where A = (Z) (7! /6 lp (1-p)15 - 32E / (3v ),

A c i r c u l a r a r c graph i s t h e i n t e r s e c t i o n graph of a f ami ly of a r c s on

a c i r c l e . Thus t h e d e f i n i t i o n of a c i r c u l a r a r c graph is t h e same a s t h a t of

an i n t e r v a l graph except t h a t " i n t e r v a l s of t h e r e a l l i n e " is r ep l aced by

" a r c s of a c i r c l e . " Tucker [18] h a s g iven a ma t r i x c h a r a c t e r i z a t i o n of

c i r c u l a r a r c graphs .

COROLLARY. Let S be t he event t h a t a random graph i s a c i r c u l a r a r c

graph. Then Theorem 3 i s t r u e when T i s r ep l aced by S.

To prove Theorem 3 from Theorem 2, we r e q u i r e a lemma which is a

c o r o l l a r y of t h e f i r s t c h a r a c t e r i z a t i o n of i n t e r v a l g raphs by Lekkerkerker

and Boland L141 g iven i n Sec t ion 2, bu t which is a l s o easy t o check d i r e c t l y .

LEMMA. I f a graph G c o n t a i n s a s an induced subgraph t h e graph I

p i c t u r e d i n F igure 1, then G is not an i n t e r v a l graph. I f G is a f o r e s t

( d i s j o i n t union of t r e e s ) and does no t con t a in I a s a subgraph, t h e n G i s an

i n t e r v a l graph.

PROOF. The first part i s obvious s i n c e I i s n o t a n i n t e r v a l graph.

For t h e second p a r t , c o n s i d e r a t r e e n o t c o n t a i n i n g I. Then it must be of

t h e form of a c a t e r p i l l a r exempl i f ied by F igure 4. To s e e t h i s , s imply l a y

Fig. 4. An example of a c a t e r p i l l a r , which i s a t r e e n o t c o n t a i n i n g a

fo rb idden subgraph of form I.

down t h e t r e e a long its l o n g e s t pa th . Now it i s e a s y t o c o n s t r u c t a s e t of

i n t e r v a l s w i t h i n t e r s e c t i o n graph cor responding t o any c a t e r p i l l a r . The

c o n s t r u c t i o n f o r t h e example i n F igure 4 i s g i v e n i n F igure 5. This p roves

t h e lemma.

F ig . 5. A s e t of i n t e r v a l s of t h e r e a l l i n e f o r which t h e i n t e r s e c t i o n

g r a p h i s t h e c a t e r p i l l a r i n F ig . 4. ,

PROOF OF THEOREM 3. Let T be t h e even t t h a t a graph does n o t c o n t a i n 1

a subgraph of t h e form of I i n F i g u r e 1. Let T be t h e even t t h a t a g raph 2

is a f o r e s t which does n o t c o n t a i n a subgraph of t h e form I. A s b e f o r e l e t

T denote t h e e v e n t t h a t a g raph is an i n t e r v a l graph. By t h e lemma, T2

i m p l i e s T which i n t u r n i m p l i e s T1. We s h a l l c a l c u l a t e t h e asympto t ic

p r o b a b i l i t y of T and show t h a t a s y m p t o t i c a l l y t h e p r o b a b i l i t y of T2 is t h e 1

same. It f o l l o w s t h a t t h e asympto t ic p r o b a b i l i t y of T e q u a l s t h e asympto t ic

p r o b a b i l i t y of T1.

Apply Theorem 2 t o t h e one-element fami ly of g r a p h s B = { G ~ ) where

G is t h e g r a p h I w i t h 7 v e r t i c e s , 6 edges and d = 12/7 p i c t u r e d i n 1

Figure 1. T is i d e n t i c a l t o t h e even t t h a t t h e g raph c o n t a i n s n o element 1

of t h e f a m i l y B. Then B1, t h e number of p o s s i b l e ways of p l a c i n g G1 = I 6 -A

on 7 l a b e l l e d v e r t i c e s , is j u s t 7* (3 )*3! = 7!/6. Thus P ( T I - e , v ,E(v) 1

where a l l t h e r e q u i r e d c o n s t a n t s i n Eq. ( 1 ) a r e now known.

To show t h a t l i m P (T ) = 0, we observe t h a t w v , ~ ( v ) ( ~ l ) - P v , ~ ( v ) 2 t h e event T1 - T is con t a ined i n t h e event C t h a t t h e graph is not a

2 f o r e s t , i . e . , i n t h e event t h a t t h e graph c o n t a i n s a c y c l e . The t h r e sho ld

func t i on f o r t h e event t h a t a random graph c o n t a i n s a cyc l e w i th k 2 3

edges is v C8, p. 231. Since we l e t l i m E/v5I6 = c and s i n c e w lim v5j6/v = 0, it is immediate t h a t l i m Pv,E(v) (C) = 0. This proves

Theorem 3.

PROOF OF COROLLARY. The lemma used i n proving Theorem 3 a p p l i e s t o

c i r c u l a r a r c graphs a s w e l l a s t o i n t e r v a l g raphs .

I n Theorem 3 , whi le A is t h e exac t expected number of subgraphs G i n a 1

random graph under t h e model r** i n which edges a r e drawn independent ly , -A v,E

e is only an approximation t o P (T) even when edges a r e drawn v,E

independently.

There is an apparen t d i sc repancy between t h e r e s u l t s of Theorems 1 and

3. According t o Theorem 3 , P (T) dec r ea se s monotonical ly a s E i n c r e a s e s , v,E

f o r any f i xed v. But F igure 6 i l l u s t r a t e s how P (TI v a r i e s over t h e whole v,E

Fig. 6. The number of pseudo-random g raphs which a r e i n t e r v a l g raphs , i n

Monte Car lo samples of 100 f o r each va lue of E and v , a s a f u n c t i o n of E/v v (upper a b s c i s s a ) o r a s a f unc t i on of E / ( ~ ) ( lower a b s c i s s a ) , where E i s t h e

number of edges and v i s t h e number of v e r t i c e s ; ( a ) v = 10, ( b ) v = 40.

These s imu la t i ons used a d i f f e r e n t pseudo-random number g e n e r a t o r from those

i n Table 1 and t h e numer ica l r e s u l t s d i f f e r s l i g h t l y . (From Cohen C71.1

THE PROBABILITY OF AN INTERVAI CRAPH, AND WHY IT MATTERS 111

range of E, f o r v = 10 (on t h e l e f t ) and v = 40 (on t h e r i g h t ) , and t h e exac t v

r e s u l t s i n Theorem 1 show t h a t a s E approaches ( 2 ) , P (T) must r i s e t o vDE

approach 1. The d iscrepancy a r i s e s because t h e asymptotic a n a l y s i s is v a l i d v only f o r va lues no t much l a r g e r than v5l6 , which vanishes r e l a t i v e t o ( ) f o r 2

l a r g e v. The methods developed he re could u s e f u l l y be app l i ed t o f i n d i n g an

asymptotic e s t ima te of P (TI f o r edges i n t h e range v5l6 < < E < (;I - g ( v ) v,E

where g ( v ) -+ a s v -+ a. It is c l e a r t h a t P (TI vanishes very r a p i d l y i n v,E

t h i s range.

To determine how l a r g e v must be f o r t h e asymptotic a n a l y s i s t o provide a

good approximation t o P (T), we e s t ima te P (T) by Theorem 3 and by Monte v,E VDE

Car lo s imu la t i on , f o r v = 10 , 40, 100, and 200, and f o r each v, f o r va lues of

E = ~ k v ~ ' ~ ] , where [ I i s t h e i n t e g e r p a r t and k = 1/3, 1/2, 2/3, 516, 1, and

4/3 (Table 1 ) .

Where E I 5, it makes l i t t l e sense t o use t h e asymptotic e s t ima te of

P (T) s i nce t h i s e s t ima te is based on t h e event T t h a t a graph does no t v,E 1

con ta in t h e graph G wi th 6 edges. For tuna te ly , exac t r e s u l t s a r e a v a i l a b l e 1

from Theorem 1 f o r E S 6. When E 1 6 , Table 1 inc ludes both t h e exac t

P (TI and asymptot ic e s t ima te s f o r comparison with t h e Monte Car lo r e s u l t s . v,E

To ob t a in t h e Monte Carlo e s t ima te s , t h e a lgor i thm of Fulkerson and

Gross C91 was programmed i n APL. ( I n f u t u r e c a l c u l a t i o n s where it is

necessary t o determine whether each of many graphs i s an i n t e r v a l graph,

t h i s a lgor i thm should be r ep l aced by t h e much f a s t e r a lgor i thm descr ibed

by Booth and Lueker C41. I n t h e i r a lgor i thm t h e number of s t e p s is l i n e a r

in v t E.) For each combination of v and E, 100 pseudo-random graphs were

genera ted . Unfor tuna te ly , t h e a lgor i thm used t o gene ra t e pseudo-random

numbers i n t h e ve r s ion of APL which we used is unknown. We co r r ec t ed t h e

shortcomings of t h e pseudo-random number gene ra to r descr ibed i n Cohen (C71,

Chapter 5 ) by changing t h e software. The Monte Car lo e s t i m a t e s of P (T) v,E

given i n Table 1 a r e t h e p ropo r t i ons of t he se genera ted graphs which A r e

i n t e r v a l graphs f o r each v and E.

I f p i s any one of t h e s e es t imated propor t ions , then an e s t ima te of

t h e s tandard dev i a t i on of p is 0 . 1 [ ~ ( 1 - ~ ) 1 ~ / ~ , which never exceeds 0.05. I t

is r ea s su r ing t h a t t h e exac t p r o b a b i l i t i e s P (T ) , where known, a r e never v,E

more than 2.5 s tandard dev i a t i ons from t h e corresponding Monte Ca r lo

e s t ima te s (MI. With v = 40 v e r t i c e s and E = 18 edges t h e standard dev i a t i on

of t h e e s t ima te p = 0.25 of P40,18(T) i s approximately 0.0433. The

asymptotic e s t ima te of 0.189 d i f f e r s from t h e Monte Car lo e s t ima te by 1 . 4

s tandard d e v i a t i o n s , which sugges t s t h a t t h e asymptotic t heo ry is u s e f u l f o r

a graph with a s few a s 40 v e r t i c e s and up t o 18 edges; f o r 21 edges, t h e

asymptotic t heo ry appears t o underes t imate P (T) r e l a t i v e t o t h e Monte v,E

Table 1. Est imates of t h e p r o b a b i l i t y P (T) t h a t a random graph wi th v l a b e l l e d v e r t i c e s and E edges i s an v*E

i n t e r v a l graph, according t o asymptotic theory ( A ) and 100 Monte- Ca r lo s imula t ions (M)

Edges E P (TI A M E A M E A M E A M v*E

v = number of v e r t i c e s

E = number of edges

P (T) = exac t p r o b a b i l i t y of an i n t e r v a l graph v*E

A = asymptotic p r o b a b i l i t y of an i n t e r v a l graph v 6 15

= e ~ p ( - ( 7 ! / 6 ) ( ~ ) p (1-p) 1, where p = E/(;)

M = Monte Ca r lo p r o b a b i l i t y of an i n t e r v a l graph based on 100 t r i a l s f o r each case

* = no t computed

~ k v ~ / ~ l = i n t e g e r p a r t of kv 5/6

# = i n t h e s e c a s e s , P(T ) = 1 when edges a r e sampled without replacement 1

C a r l o e s t i m a t e s . For v = 100 v e r t i c e s , a l l t h e asympto t ic e s t i m a t e s a r e

w i t h i n 2.5 s t a n d a r d d e v i a t i o n s o f t h e cor responding Monte C a r l o e s t i m a t e s ;

f o r v = 200, t h e r e a r e some l a r g e r d e v i a t i o n s . Overa l l , we conclude t h a t t h e

asympto t ic t h e o r y is u s e f u l f o r g r a p h s w i t h 100 or mare v e r t i c e s , a s long a s

t h e number of edges does n o t g r e a t l y exceed v 5 l 6 . C a l c u l a t i o n s n o t shown 6 5

here of t h e asympto t ic p r o b a b i l i t y u s i n g exp(-(32/3)(E /v ) ) approximate t h e

Monte C a r l o e s t i m a t e s w i t h i n 3 .2 s t a n d a r d d e v i a t i o n s i n a l l c a s e s when

v = 200, though no t s o w e l l w i t h s m a l l e r v a l u e s of v. Thus A* is u s e f u l f o r

v L 200.

The asympto t ic e s t i m a t e s i n Theorem 3 could probably be improved by

t a k i n g i n t o account c y c l e s a s f o r b i d d e n induced subgraphs. However, t h e

r e s u l t s would be more compl ica ted and would n o l o n g e r a p p l y t o c i r c u l a r a r c

g raphs , which may have c y c l e s a s induced subgraphs.

Benzer [l] observed E = 6 1 (which i s one h a l f t h e number of

o f f - d i a g o n a l 0 ' s i n h i s F i g u r e 5 ) w i t h v = 19. Using h i n Theorem 3, t h e

'%symptotic e s t i m a t e of P (T) i s fl. From t h e r a t e of d e c r e a s e of t h e 1 9 , 6 1 5 /6 Monte C a r l o e s t i m a t e s of PIO,E(T) i n Table 1 a s E i n c r e a s e s beyond 1 0 ,

and because E = 6 1 is much l e s s t h a n ( l ; ) = 171, it a p p e a r s l i k e l y t h a t t h e

chance t h a t Benzer observed an i n t e r v a l g raph by chance a l o n e is n e a r l y 0.

5. UNSOLVED PROBLEMS. There remain unsolved mathemat ica l problems

r e l a t e d t o random i n t e r v a l g raphs . S o l u t i o n s would be u s e f u l i n s c i e n t i f i c

i n f e r e n c e .

F i r s t , t h e problem of c a l c u l a t i n g P (T) i s r e a l l y a s p e c i a l c a s e v,E

of c a l c u l a t i n g t h e p r o b a b i l i t y d i s t r i b u t i o n of t h e minimum dimension of

Eucl idean space necessary t o r e p r e s e n t a random graph a s t h e i n t e r s e c t i o n

graph of a f a m i l y of s e t s o f some g i v e n f o r m . For example, t h e minimum

number of dimensions n e c e s s a r y t o r e p r e s e n t a g raph G by t h e i n t e r s e c t i o n s

of boxes, or r e c t a n g u l a r p a r a l l e l i p i p e d s w i t h edges p a r a l l e l t o t h e axes ,

is c a l l e d t h e b o x i c i t y of G, and never exceeds [v/21 (Rober t s C151). If

[B = k l is t h e e v e n t t h a t t h e b o x i c i t y of a g i v e n g r a p h i s k, [B = 01 i s

t h e even t t h a t t h e g raph i s a complete g raph , s i n c e a l l v e r t i c e s c a n be

r e p r e s e n t e d by c o i n c i d e n t p o i n t s , and T = [B = 0 ar B = 11. It would be

u s e f u l t o know t h e p r o b a b i l i t y P [B = k l t h a t a random graph is of v,E

b o x i c i t y k , k = 0, 1, . . . , [v/21. For o t h e r f a m i l i e s of s e t s , such

a s convex s e t s , t h e problem may b e e a s i e r ; e v e r y graph is t h e i n t e r s e c t i o n

graph of convex s e t s i n 3 o r fewer dimensions (Wegner 1191).

Second, t h e r e i s s t i l l n o t h e o r y f o r t h e p r o b a b i l i t y of a n i n t e r v a l

g raph when t h e p r o b a b i l i t y d i s t r i b u t i o n i s d e f i n e d i n i t i a l l y on t h e d i r e c t e d

graphs cor responding t o food webs i n ecology, r a t h e r t h a n on t h e u n d i r e c t e d

n iche o v e r l a p g r a p h s a s i n t h e t h e o r y of Erdlis and ~ 6 n ~ i C81. Far example,

114 JOEL E. COHEN, J ~ N O S K O M L ~ S , AND THOMAS MUELLER

in ecology it a p p e a r s (Cohen [7]) t h a t a u s e f u l model of a d i r e c t e d graph

W is a s fo l lows: g i v e n a s e t of v l a b e l l e d v e r t i c e s (cor responding t o v

p r e d a t o r s ) which may have p o s i t i v e in-degree, and a n o t n e c e s s a r i l y

d i s j o i n t s e t of u l a b e l l e d v e r t i c e s (cor responding t o u p r e y ) which may

have p o s i t i v e ou t -degree , an arrow from a p r e y v e r t e x t o a p r e d a t o r v e r t e x

in W a c t u a l l y occurs wi th p r o b a b i l i t y p , and f a i l s t o occur wi th p r o b a b i l i t y

1-p, independent ly and w i t h i d e n t i c a l p r o b a b i l i t y for each p a i r of p r e y and

p r e d a t o r v e r t i c e s . From each such random d i r e c t e d graph W, t h e random

u n d i r e c t e d graph G(W) on t h e v ( p r e d a t o r ) v e r t i c e s j o i n s two v e r t i c e s

wi th a n u n d i r e c t e d edge i f and o n l y i f t h e r e is a prey v e r t e x i n W from

which arrows g o t o b o t h p r e d a t o r v e r t i c e s . The problem is t o f i n d t h e

p r o b a b i l i t y t h a t G(W) i s a n i n t e r v a l g raph , o r b e t t e r , t h e p r o b a b i l i t y

d i s t r i b u t i o n of t h e bcv t ic i ty o f G .

ACKNOWLEDGMENTS. J.E.C. t h a n k s members of t h e Mathematical I n s t i t u t e

of t h e Hungarian Academy of S c i e n c e s for k ind h o s p i t a l i t y d u r i n g a v i s i t i n

1973. The v i s i t , under t h e a u s p i c e s of an exchange program between t h e

Hungarian Academy o f Sc iences and t h e N a t i o n a l Academy of S c i e n c e s o f t h e

U.S.A., l e d t o t h i s c o l l a b o r a t i v e work. William T. T r o t t e r , Jr. c o r r e c t e d

s l i p s i n an e a r l i e r v e r s i o n . Anne Whit taker p repared t h i s t y p e s c r i p t .

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THE ROCKEFELLER UNIVERSITY, 1230 YORK AVENUE, NEW YORK, NY 10021, U.S.A.

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